A steel ball is dropped from th roof of a building. `A` man standing in front of a `1-m` high window in the building notes tha the ball takes `0.1 s` to the fall from the top to the boottom of the window. The ball continues to fall and strikes the ground. On striking the ground, the ball gers rebounded with the same speed with which it hits the ground. If the ball reappears at the bottom of the window `2 s` after passing the bottom of the window on the way down, dind the height of the building.

Explanation

To solve the problem step by step, we will analyze the motion of the steel ball as it falls and rebounds.

### Step 1: Understand the motion of the ball
The ball is dropped from the roof of a building and falls through a 1-meter high window in 0.1 seconds. After passing the bottom of the window, it continues to fall until it hits the ground and then rebounds with the same speed.

### Step 2: Calculate the speed of the ball at the bottom of the window
Using the equation of motion:
\[ s = ut + \frac{1}{2}gt^2 \]
where:
- \( s = 1 \, \text{m} \) (height of the window)
- \( u = 0 \, \text{m/s} \) (initial velocity when dropped)
- \( t = 0.1 \, \text{s} \) (time taken to fall through the window)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

Substituting the values:
\[ 1 = 0 \cdot 0.1 + \frac{1}{2} \cdot 9.8 \cdot (0.1)^2 \]
\[ 1 = \frac{1}{2} \cdot 9.8 \cdot 0.01 \]
\[ 1 = 0.049 \, \text{m} \]
This confirms that the ball falls 1 meter in 0.1 seconds.

### Step 3: Calculate the velocity of the ball at the bottom of the window
Using the equation:
\[ v = u + gt \]
Substituting the values:
\[ v = 0 + 9.8 \cdot 0.1 = 0.98 \, \text{m/s} \]
The velocity at the bottom of the window is approximately \( 0.98 \, \text{m/s} \).

### Step 4: Determine the time taken to fall from the bottom of the window to the ground
After passing the bottom of the window, the ball takes an additional 2 seconds to reappear at the bottom of the window after hitting the ground. Since it takes 1 second to fall to the ground and 1 second to rebound back to the bottom of the window, we can analyze the fall.

### Step 5: Calculate the height of the building
The total time taken to fall from the top of the building to the ground can be expressed as:
\[ t_{\text{total}} = t_{\text{window}} + t_{\text{fall}} + t_{\text{rebound}} = 0.1 + 1 + 1 = 2.1 \, \text{s} \]

Using the equation of motion to find the height of the building \( H \):
\[ H = \frac{1}{2} g t_{\text{total}}^2 \]
Substituting the values:
\[ H = \frac{1}{2} \cdot 9.8 \cdot (2.1)^2 \]
Calculating:
\[ H = \frac{1}{2} \cdot 9.8 \cdot 4.41 \approx 21.56 \, \text{m} \]

### Final Answer
The height of the building is approximately \( 21.56 \, \text{m} \).

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